The Milne problem in a continuous stochastic planar medium with Gaussian statistics

The Milne problem of radiative transfer in a planar medium, with isotropic scattering is considered. The medium is assumed to be continuous stochastic medium, with fluctuations described as Gaussian field. Pomraning-Eddington method is used to obtain an explicit form for the radiation energy density in the deterministic case. It depends on two random variables, namely the optical space variable and the optical thickness of the medium. The Gaussian joint probability density function of these two random variables is defined and used to find the ensemble-averaged energy density and the linear extrapolation distance. It is shown that the statistical nature of the medium leads to two quite different solutions of the Milne problem. Numerical results are implemented for the sake of clarification.